2. CompSci 100e 7.1 Plan for the week l Review:  Union-Find l Understand linked lists from the bottom up and top-down  As clients of java.util.LinkedList  As POJOs  Using linked lists to leverage algorithmic improvements l Binary Trees  Just a linked list with 2 pointers per node? l Self-referential structures and recursion  Why recursion works well with linked-structures l Lab: Boggle  Testing & Analysis

3. CompSci 100e 7.2 Contrast LinkedList and ArrayList l ArrayList --- why is access O(1) or constant time?  Storage in memory is contiguous, all elements same size  Where is the 1 st element? 40 th ? 360 th ?  Doesn’t matter what’s in the ArrayList, everything is a pointer or a reference (what about null?)

4. CompSci 100e 7.3 What about LinkedList? l Why is access of N th element linear time? l Why is adding to front constant-time O(1)? front

5. CompSci 100e 7.4 Linked list applications continued l If programming in C, there are no “growable-arrays”, so typically linked lists used when # elements in a collection varies, isn’t known, can’t be fixed at compile time  Could grow array, potentially expensive/wasteful especially if # elements is small.  Also need # elements in array, requires extra parameter  With linked list, one pointer used to access all the elements in a collection l Simulation/modeling of DNA gene-splicing  Given list of millions of CGTA… for DNA strand, find locations where new DNA/gene can be spliced in Remove target sequence, insert new sequence

6. CompSci 100e 7.5 Linked lists, CDT and ADT l As an ADT  A list is empty, or contains an element and a list  ( ) or (x, (y, ( ) ) ) l As a picture l As a CDT (concrete data type) pojo: plain old Java object public class Node { Node p = new Node(); String info; p.info = “hello”; Node next; p.next = null; } p 0

7. CompSci 100e 7.6 Building linked lists l Add words to the front of a list (draw a picture)  Create new node with next pointing to list, reset start of list public class Node { String value; Node next; Node(String s, Node link){ value = s; next = link; } // … declarations here Node list = null; while (scanner.hasNext()) { list = new Node(scanner.next(), list); } l What about adding to the end of the list?

8. CompSci 100e 7.7 Dissection of add-to-front l List initially empty l First node has first word l Each new word causes new node to be created  New node added to front l Rhs of operator = completely evaluated before assignment list A list = new Node(word,list); B Node(String s, Node link) { info = s; next = link;} list

9. CompSci 100e 7.8 Standard list processing (iterative) l Visit all nodes once, e.g., count them or process them public int size(Node list){ int count = 0; while (list != null) { count++; list = list.next; } return count; } l What changes in code if we generalize what process means?  Print nodes?  Append “s” to all strings in list?

10. CompSci 100e 7.9 Standard list processing (recursive) l Visit all nodes once, e.g., count them public int recsize(Node list) { if (list == null) return 0; return 1 + recsize(list.next); } l Base case is almost always empty list: null pointer  Must return correct value, perform correct action  Recursive calls use this value/state to anchor recursion  Sometimes one node list also used, two “base” cases l Recursive calls make progress towards base case  Almost always using list.next as argument

11. CompSci 100e 7.10 Recursion with pictures l Counting recursively int recsize(Node list){ if (list == null) return 0; return 1 + recsize(list.next); } recsize(Node list) return 1+ recsize(list.next) recsize(Node list) return 1+ recsize(list.next) recsize(Node list) return 1+ recsize(list.next) recsize(Node list) return 1+ recsize(list.next) System.out.println(recsize(ptr)); ptr

12. CompSci 100e 7.11 Recursion and linked lists l Print nodes in reverse order  Print all but first node and… Print first node before or after other printing? public void print(Node list) { if (list != null) { } print(list.next); System.out.println(list.info); print(list.next);

13. CompSci 100e 7.12 Binary Trees l Linked lists: efficient insertion/deletion, inefficient search  ArrayList: search can be efficient, insertion/deletion not l Binary trees: efficient insertion, deletion, and search  trees used in many contexts, not just for searching, e.g., expression trees  search in O(log n) like sorted array  insertion/deletion O(1) like list, once location found!  binary trees are inherently recursive, difficult to process trees non-recursively, but possible recursion never required, often makes coding simpler

14. CompSci 100e 7.13 From doubly-linked lists to binary trees l Instead of using prev and next to point to a linear arrangement, use them to divide the universe in half  Similar to binary search, everything less goes left, everything greater goes right  How do we search?  How do we insert? “llama” “tiger” “monkey” “jaguar” “elephant” “ giraffe ” “pig” “hippo” “leopard” “koala”

15. CompSci 100e 7.14 Basic tree definitions l Binary tree is a structure:  empty  root node with left and right subtrees l terminology: parent, children, leaf node, internal node, depth, height, path link from node N to M then N is parent of M –M is child of N leaf node has no children –internal node has 1 or 2 children path is sequence of nodes, N 1, N 2, … N k –N i is parent of N i+1 –sometimes edge instead of node depth (level) of node: length of root-to-node path –level of root is 1 (measured in nodes) height of node: length of longest node-to-leaf path –height of tree is height of root l Trees can have many shapes: short/bushy, long/stringy  If height is h, how many nodes in tree? A B ED F C G

16. CompSci 100e 7.15 A TreeNode by any other name… l What does this look like?  What does the picture look like? public class TreeNode { TreeNode left; TreeNode right; String info; TreeNode(String s, TreeNode llink, TreeNode rlink){ info = s; left = llink; right = rlink; } “llama” “tiger” “ giraffe ”

17. CompSci 100e 7.16 Printing a search tree in order l When is root printed?  After left subtree, before right subtree. void visit(TreeNode t) { if (t != null) { visit(t.left); System.out.println(t.info); visit(t.right); } l Inorder traversal l Big-Oh? “llama” “tiger” “monkey” “jaguar” “elephant” “ giraffe ” “pig” “hippo” “leopard”

18. CompSci 100e 7.17 Tree traversals l Different traversals useful in different contexts  Inorder prints search tree in order Visit left-subtree, process root, visit right-subtree  Preorder useful for reading/writing trees Process root, visit left-subtree, visit right-subtree  Postorder useful for destroying trees Visit left-subtree, visit right-subtree, process root “llama” “tiger” “monkey” “jaguar” “elephant” “ giraffe ”

19. CompSci 100e 7.18 Tree functions l Compute height of a tree, what is complexity? int height(Tree root) { if (root == null) return 0; else { return 1 + Math.max(height(root.left), height(root.right) ); } l Modify function to compute number of nodes in a tree, does complexity change?  What about computing number of leaf nodes?

20. CompSci 100e 7.19 Balanced Trees and Complexity l A tree is height-balanced if  Left and right subtrees are height-balanced  Left and right heights differ by at most one boolean isBalanced(Tree root) { if (root == null) return true; return isBalanced(root.left) && isBalanced(root.right) && Math.abs(height(root.left) – height(root.right)) <= 1; }

21. CompSci 100e 7.20 What is complexity? l Assume trees are “balanced” in analyzing complexity  Roughly half the nodes in each subtree  Leads to easier analysis l How to develop recurrence relation?  What is T(n)?  What other work is done? l How to solve recurrence relation  Plug, expand, plug, expand, find pattern  A real proof requires induction to verify correctness